I won't tell you what it cost to fix our "muddy run" problem, but I do admit I love seeing the girls out in the run now and not getting their feet all muddy and right now full of snow. We live in central NY and we have lots of snow now and rain in the spring. Last year we tried tarps, forget that, so this summer we spent lots of $ and time, but they have a 10x30' roofed run. there is a 10x20' extension, but no roof. They won't even go near that unroofed partnow that there is snow in it, but I do love seeing them out side and comfortable under cover. Its so much easier for cleaning also. Last year I felt like you and dreaded going out in that gross run, but I do love the visits this year. I hope you can find a less expensive solution, but it is $ well spent for us. Sand will still get wet, as you know, but less mud if you put it deep enough. Good luck no matter what you do.
Oh, one more thing, we do put clear plastic around all the outer walls so the snow doesn't blow in. And it's sheltered from the wind with the plastic sides.